# A triangle has sides A, B, and C. The angle between sides A and B is pi/4 and the angle between sides B and C is pi/12. If side B has a length of 42, what is the area of the triangle?

Dec 3, 2017

Area of $\Delta = 187.5281$

#### Explanation:

$\angle A = \frac{\pi}{12} , \angle C = \frac{\pi}{4} , \angle B = \left(\pi - \left(\left(\frac{\pi}{12}\right) + \left(\frac{\pi}{4}\right)\right)\right) = \frac{2 \pi}{3}$

$\frac{A}{\sin} A = \frac{B}{\sin} B = \frac{C}{\sin} C$

$\frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{42}{\sin} \left(\frac{2 \pi}{3}\right) = \frac{C}{\sin} \left(\frac{\pi}{4}\right)$

$A = \frac{42 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{2 \pi}{3}\right) = \frac{42 \cdot 9.2588}{0.866} = 12.5515$

$C = \frac{42 \cdot \sin \left(\frac{\pi}{4}\right)}{\sin} \left(\frac{2 \pi}{3}\right) = \frac{42 \cdot 0.707}{0.866} = 34.2887$

Semi perimeter $s = \frac{A + B + C}{2} = \frac{12.5515 + 42 + 34.2887}{2}$
$s = 44.4201$

$s - A = 44.4201 - 12.5515 = 31.8686$
$s - B = 44.4201 - 42 = 2.4201$
$s - C = 44.4201 - 34.2887 = 10.1314$

Area of $\Delta = \sqrt{s \left(s - A\right) \left(s - B\right) \left(s - C\right)}$
Area of $\Delta = \sqrt{44.4201 \cdot 31.8686 \cdot 2.4201 \cdot 10.1314}$
Area of $\Delta = \sqrt{35166.8011} = 187.5281$